16 ANALYTICAL MECHANICS
while the book presses downward upon your hand. The law states that the action of your hand equals the reaction of the book and is in the opposite direction. The book reacts upon your hand because the earth attracts it. When your hand and the earth are the only bodies which act upon the book, the action of your hand equals and is opposite to the action of the earth. In other words the sum of the two actions is nil. Generalizing from this simple illustration we can put the law into the following form:
To every action there is an equal and opposite reaction, or the sum of all the actions to which a body or a part of a body is subject at any instant vanishes :
SA = 0.* (A)
25. Condition for the Equilibrium of a Particle. — The condition of equilibrium of a particle is obtained by replacing the term "action" by the term "force" in the last form of the fundamental law and then stating it in the form of a condition. Thus — in order that a particle be in equilibrium the sum of all the forces which act upon it must vanish.
In other words if Fi, F2, F3, . . . , Fn are the forces winch act upon a particle, then the vector equation
Fi+F2+F3+- - - + Fn = 0 (I)
must be satisfied in order that the particle be in equilibrium. Equation (I) is equivalent to stating that when the forces are added graphically they form a closed polygon. But when the sum of a number of vectors vanishes the sum of their components also vanishes. Therefore we must have
X1+X2+- • - +Xn=0,
n ;,
where Xt; Y,, and Zt are the components of F<.* Since the vectors in each of the equations of (IF) are parallel we can
* See table of notations.